It's hard to imagine any sentient creature turning down the chance of sharing their TARDIS with Karen Gillan – even if they're not quite sure what a TARDIS is. The vivacious 24-year-old Scottish redhead is the breakout star of the wildly addictive current incarnation of the British sci-fi franchise Doctor Who – which tells the tale of the Doctor, an immortal, shape-shifting alien who travels through space and time via the TARDIS (a time machine that's bigger on the inside than the outside). But why does he always bring a cute Earth girl along for the ride? "Well, why wouldn't he?" says Gillan, who plays Amy Pond, the doctor's current, micro-mini-wearing companion.

Doctor Who has been around since 1963, but its latest version is the best sci-fi show on the planet, thanks to a hot young cast (the current Doctor is played by 29-year-old floppy-haired heartthrob Matt Smith) and mad-genius showrunner Steven Moffat: It's the kind of show where an ominous crack in the wall in a little girl's room turns out to be a deadly fracture in the fabric of space and time.

With BBC America pushing the show hard (it airs Saturdays at 9 p.m./8c), it's finally catching on here, too. When Gillan showed up at the San Diego Comic-Con this year, the place practically combusted: "There were just so many screaming, crying people," says Gillan. "I feel like I've joined the whole sort of nerd-power world. I used to watch The X-Files and Star Trek, so I always had the potential, but I wasn't a full geek until I joined this show, and now I've become a full-on geek." (She also notes that some of the male attendees "wouldn’t look me directly in the eye, which is quite funny.")

As Amy, Gillan has faced off with voracious aliens and Adolf Hitler, played a 65-year-old version of herself, and watched the entire universe reboot itself at least twice. "It's like no other acting job," she says. But now she's leaving midway through the current season. "I didn't want to overstay my welcome," she says, "or let it fizzle out in any way. I wanted to leave with a bang, so I thought, now's the time to do it. I just had this gut instinct."

Now Gillan is headed for Hollywood, beginning with a horror movie she'll shoot in Alabama: "I just want to eat some hamburgers there," she says. But she's already missing the world of the Doctor. "We wrapped about a month ago, so it's all done, which is a very strange sensation," she says. "It feels really weird, because the last three years of my life, like, every day, pretty much, have been running away from monsters. To not be doing that anymore is a bit strange. I'm like, 'What, I have to do acting jobs where I'm just talking to people in a room? What is this? Where's the alien?'"